Article
Order Reconstruction in a Nanoconfined Nematic Liquid Crystal between Two Coaxial Cylinders Xuan Zhou, Zhidong Zhang *, Qian Zhang and Wenjiang Ye Received: 8 October 2015; Accepted: 16 November 2015; Published: 30 November 2015 Academic Editor: Ingo Dierking Department of Physics, Hebei University of Technology, Tianjin 300401, China; [email protected] (X.Z.); [email protected] (Q.Z.); [email protected] (W.Y.) * Correspondence: [email protected]; Tel.: +86-22-6043-5589; Fax: +86-22-6043-8055
Abstract: The dynamics of a disclination loop (s = ˘1/2) in nematic liquid crystals constrained between two coaxial cylinders were investigated based on two-dimensional Landau–de Gennes tensorial formalism by using a finite-difference iterative method. The effect of thickness (d = R2 ´ R1 , where R1 and R2 represent the internal and external radii of the cylindrical cavity, respectively) on the director distribution of the defect was simulated using different R1 values. The results show that the order reconstruction occurs at a critical value of dc , which decreases with increasing inner ratio R1 . The loop also shrinks, and the defect center deviates from the middle of the system, which is a non-planar structure. The deviation decreases with decreasing d or increasing R1 , implying that the system tends to be a planar cell. Two models were then established to analyze the combined effect of non-planar geometry and electric field. The common action of these parameters facilitates order reconstruction, whereas their opposite action complicates the process. Keywords: liquid crystal; order reconstruction; disclination loop; cylindrical wall; biaxial transition; Landau–de Gennes theory
1. Introduction The equilibrium configuration of a confined nematic liquid crystal (NLC) is caused by interplay among surface interaction, elastic distortion, and finite-size effect. Confined systems that suffer from continuous symmetry-breaking transitions often display topological defects [1–4], which are utilized in new-generation LC devices. Therefore, study of defects is an important field in physics. In particular, defects in nanoconfined LC cells have received increased research attention. A weak local perturbation in an LC confined in a nanoscale cell can stimulate an apparent mesoscopic or even macroscopic response [5]. Eigenvalue exchange/order reconstruction occurs under severe confinement in the presence of antagonistic anchorings, and this mechanism relaxes surface-induced frustrations. Order reconstruction structures have been investigated in detail [6–11] for various boundary conditions in nematic cells bounded by parallel walls, for which the characteristic linear size of the confining plates is large and virtually infinite compared with the cell thickness. These structures are stable in a sufficiently thin cell [6,7], whose gap is comparable with the order parameter length ξ0 (characteristic length for order-parameter changes), or when a sufficiently strong electric or magnetic field is applied [8,9]. Studies [12–14] revealed that mesoscopic Landau–de Gennes theory can accurately predict the structural and phase behavior of severely confined LCs. Although LC defects have been intensively studied theoretically and experimentally, LC shells have been rarely investigated. LC shells have received increased research interest because of their potential to generate colloids with a valence, which can be used to build colloidal architectures for
Materials 2015, 8, 8072–8086; doi:10.3390/ma8125446
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Materials 2015, 8, 8072–8086
photonic applications. Several scholars have numerically simulated the possible defect configurations in nematic and smectic shells [15–17]. NLCs in a cylindrical cavity have been studied for approximately 40 years. Cylindrical geometry is the most convenient geometry after planar cell geometry for theoretical and experimental analyses. Ñ As such, all experimentally observed and theoretically possible configurations of the director field n yield a complete or approximate analytic description [18]. The possible equilibrium configurations of Ñ an NLC in a cylindrical cavity primarily depend on how n is anchored to the cylinder surface. In this study, we investigated the structural behavior of a nanoscale NLC system confined in a cylindrical cavity with a topological defect loop by using the finite-difference iterative method. The outline of the paper is as follows. In Section 2, we introduce the phenomenological model employed and describe the geometry of the problem and our parametrization. The results are presented in Section 3, and the conclusions are summarized in Section 4. 2. Theoretical Basis 2.1. Free Energy Our theoretical argument is based on Landau–de Gennes theory [19], wherein the orientational order of LC is described by a second-rank symmetric and traceless tensor [3]: Q“
3 ÿ
Ñ
Ñ
λi e i b e i
(1)
i “1
Ñ
where e i is the orthogonal unit vector the eigenvector of Q, and λi is its eigenvalue. ˆ representing ˙ 1 2 The range of eigenvalues must be λi P ´ , to interpret Q as the traceless second-moment tensor 3 3 of the molecular distribution function. Q vanishes in the isotropic phase, whereas this parameter has two degenerate eigenvalues in the uniaxial ordering and can be represented by ˆ Q“S
1 nbn´ I 3
Ñ
Ñ
˙ (2)
Ñ
where n is the nematic director pointing along the local uniaxial ordering direction, and S is the uniaxial scalar parameter expressing the magnitude of fluctuations about the nematic director. In Equation (2), S can be either positive or negative. The ensemble of molecules represented by Q Ñ Ñ tends to align along n when S is positive and tends to lie in the plane orthogonal to n when S is negative. Finally, the LC is in a biaxial state when all the eigenvalues of Q are distinct. The degree of biaxiality is expressed by the parameter β2 defined as [20] “ ` ˘‰2 6 tr Q3 β “ 1 ´ “ ` ˘‰3 , tr Q2 2
(3)
This equation is a convenient parameter for illustrating spatial inhomogeneities of Q and ranges in the interval [0,1]. All uniaxial states with two degenerate eigenvalues correspond to β2 = 0, whereas states with maximal biaxiality correspond to β2 = 1. As tr(Q3 ) = 3detQ, states with β2 = 1 are those with det Q = 0, which implies that at least one eigenvalue of Q vanishes. The Landau–de Gennes free energy density of LC is given by f = fbulk {Qαβ } + felastic {Qαβ ,5} + fdielectric {Qαβ }, in which ¯2 A B C´ (4) f bulk “ trQ2 ´ trQ3 ` trQ2 2 3 4
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is the bulk energy that describes a homogeneous phase. In fbulk , B and C are constants, and A is assumed to vary with temperature T in the form of A = a (T ´ T*), where a is a positive constant and T* is the nematic supercooling temperature. Equation (4) the bulk ˜ provides ¸ equilibrium value of c 24AC B 1+ 1 ´ , which depends on the the uniaxial scalar order parameter in Equation (2) Seq = 4C B2 temperature. The free-energy felastic , which penalizes gradients in the tensor order parameter field, is given in the form 1 f elastic “ L |∇ Q|2 (5) 2 where the one-elastic-constant approximation is used for simplicity, and the elasticity of the system is represented by a single positive elastic constant L. The dielectric contribution to the free energy density is given by [21] f dielectric “ ´
ε0 Ñ Ñ E ¨ εE 2
(6)
where ε0 is the vacuum electric permeability constant, and ε is the dielectric tensor to describe the Ñ
local anisotropic response of the nematic ordering to E . ε is generally expressed as ε “ εi I ` ε a Q
(7)
where εi = (εk + 2εK )/3 and εa = (εk ´ εK )/Seq are isotropic and anisotropic dielectric susceptibilities, Ñ
respectively. When εa > 0, similar to the following calculation, nematic uniaxial ordering is along E and therefore fdielectric can be rewritten as f dielectric “ ´
ε0 2
ˆ
Ñ2
Ñ
Ñ
˙
εi E ` ε a E ¨ Q E
(8)
2.2. Geometry of the Problem Let us consider an NLC confined between two coaxial cylindrical surfaces whose internal and external radii are R1 and R2 , and the thickness of the shell is d = R2 ´ ´ R1 (Figure 1a). ¯ The standard polar cylindrical coordinates pr, ϑ, zq and the corresponding local frame Ñ
Ñ Ñ Ñ e r , e ϑ, e z
are introduced,
Ñ
where e z points along the symmetry axis, e r is the radial unit vector emanating from the symmetry Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
axis, and e ϑ :“ e z ˆ e r . The electric field E is applied along the e r axis direction, i.e., E “ E e r , Ñ which promotes the alignment of the nematic director along the e r axis. LC molecules on the surfaces of internal and external cylinders are strongly anchored along the parallel and perpendicular directions, respectively. According to the investigations of Cavallaro et al. [22], we make the assumption that the LC texture exhibits a cylindrical symmetry along the cylindrical axis, i.e., the nematic orientation is independent of ϑ. By varying the polar angle Ñ θ1 on the internal surface, where θ1 is the angle with respect to e z in the plane ϑ-z, we have verified that the choice of θ1 = 0 has the minimum energy, i.e., the director field has no azimuthal component and occurs in the plane r-z. This is a realistic choice of boundary condition for a cylindrical surface in industrial production. Generally, a homogenous planar alignment is achieved firstly by rubbing on flexible substrate, and then the substrate is fit on the solid cylinder, with the rubbing direction along the symmetric axis. It follows that the texture can be discussed in terms of 2D nematics corresponding Ñ to each radial slice. In the plane r-z, the nematic director is given by n “ psinβ, 0, cosβq, where â is Ñ the angle with respect to e z (as shown in Figure 1) and the order tensor Q can be represented as
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n = ( sin β,0,cos β) , where â is the angle with respect to ez
(as shown in Figure 1) and the order
tensor Q can be represented as Materials 2015, 8, 8072–8086
Hence,
eϑ
0 Qrz Qrr Q ( r , z ) = 0 Qϑϑ 0 , ¨ ˛ Q Q 0 Q 0 Q rr rz zz ‹ ˚ rz
(9)
Q pr, zq “ ˝
(9) 0 Qϑϑ 0 ‚, Q 0 Q zz is always an eigenvector of Q.rzThis configuration rules out distortions twisted
Ñ always an(9)eigenvector of Q.departures This configuration out distortions eϑHence, e1 , e2twisted , e3 ) of . Qrz einϑ isEquation reveals the of the rules eigenframe ( ´ ¯ along Ñe ϑ. Qrz in Equation (9) reveals the departures of the eigenframe Ñe 1 , Ñe 2 , Ñe 3 of Q from er , eÑϑ , eÑz )¯, with both frames coinciding when Qrz = 0. from´(Ñ
along
Q
e r , e ϑ , e z , with both frames coinciding when Qrz = 0.
We denote the free boundary conditions at the upper and lower lateral walls z = ±dz/2, with We denote the free boundary conditions at the upper and lower lateral walls z = ˘dz /2, with initialinitial uniaxial ordering −π/2(z(z dz/2)and and +π/2 = z−d z/2). These boundary uniaxial orderingby bytotal total rotations rotations ofof´π/2 = d=z /2) +π/2 (z =(z´d /2). These boundary conditions are are consistent with of aa defect defectloop loopwith with topological charge s = −1/2. conditions consistent withthe thegeneration generation of topological charge s = ´1/2. Figure 1b shows the director profile in a cross-section along an arbitrary radius of the system. Figure 1b shows the director profile in a cross-section along an arbitrary radius of the system. The Theare results also applicable to defect the defect loop withtopological topological charge with exchanged results alsoare applicable to the loop with charges = s 1/2, = 1/2, with exchanged conditions of the upper and lower walls. conditions of the upper and lower walls.
Figure 1. Geometry of the problem. cavityfilled filledwith with NLCs; (b) the Figure 1. Geometry of the problem.(a)(a)Sketch Sketchof of cylindrical cylindrical cavity NLCs; andand (b) the director profile in a cross-section along an arbitrary radius of the system. director profile in a cross-section along an arbitrary radius of the system.
2.3. Scaling and and Dimensionless Evolution 2.3. Scaling Dimensionless EvolutionEquations Equations «
ff
4 r ijB” Qij {q0 ,r introducedthe the following quantities: fr ” f { f ≡ f3 , Q We We introduced followingdimensionless dimensionless quantities: , Qzij ”≡z{ξ Qij0 , q0 , p4Cq 3
B4
b (4C ) r i ε0 , E r “ E{E0 , where q0 “ B{4C is the superheating 2L{ Aε c q = B 4C z ≡ z ξ0 , r ≡ r ξ0 , ε a = εa q0 εi , E0 = q0 ξ0 2 L A εi ε 0 , E = E dE0L, where 4CL0 order parameter at the nematic superheating temperature T**, and ξ0 “ “ is the B2 T**, and is the superheating order parameter at the nematic superheating Bq temperature 0 characteristic length for order-parameter changes. Equations (4), (5), and (8) can be reduced to r r ” r{ξ0 , r εa “ εa q0 {εi , E0 “ pq0 {ξ0 q
ξ0 =
L = Bq0
(
)
4CL is the characteristic length for order-parameter changes. Equations (4), (5), r B2 A 1 ´ r2 ¯2 r r2 1 r3
and (8) can be reduced to
tr Q ´ tr Q ` tr Q 3 16 1 ˇˇ r rˇˇ2 A frelastic “ 1 2 ˇ∇ 3Q+ˇ 1 trQ 2 = trQ 2 − trQ ı 12 16 13” r2 r rr E r2 E `r εa Q frdielectric “ ´ r A
f bulk “
fbulk
(10)
12
(
1 felastic = ∇ Q 2
4
2
(11)
(10)
(12)
2
1 fdielectric = − E 2 + ε a Q rr E 2 A 8075
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(11)
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r “ 24AC defines the temperature scale. Isotropic–nematic transition where the reduced parameter A B2 r “ 8{9. In polar cylindrical coordinates, Equation (11) can be expressed precisely by occurs at A »˜
fre “
¸2 ˜ ¸2 ˜ ¸2 ˜ ¸2 ˜ ¸2 ˜ ¸2 fi r r r r r r 1 – B Qrr B Qrr B Qϑϑ B Qϑϑ B Qzz B Qzz fl ` ` ` ` ` 2 Br r Br z Br r Br z Br r Br z »˜ ¸2 ˜ ¸2 fi ´ ¯ r rz r rz BQ BQ r2 ` Q r2 ` Q r2 ´ 2Q r rr Q rϑϑ . fl ` 1 Q ` `– rr rz ϑϑ r Br r Br z r2
(13)
Thus, the reduced uniaxial ordering has the form 2
Ñ r “ Sp3 r Ñ Q n b n ´ Iq ,
(14)
a Seq r is the reduced uniaxial scalar parameter at equilibrium. “ 1` 1´ A q0 We compute the evolution of LC with dynamic theory for tensor order-parameter field Q(r,z,t). Ñ The local values of the scalar-order parameter S and the director n can be calculated from Q by using the highest eigenvalue and the associated eigenvector, respectively. According to [23], the evolution equation describing the dynamics of Q can be written as where r s“
„ ˆ ˙ δf 1 δf BQ “Γ ´ ` tr I , Bt δQ 3 δQ
(15)
where Γ = 6D*/[1 ´ 3tr(Q2 )]2 , D* is the rotational diffusion for the nematic, and δ f {δQ is assumed to be symmetrical. Numerical calculations are performed using the reduced variables. When the functional r can be obtained. derivatives in Equation (15) are evaluated, the following evolution equations for Q « ˜ ¸ ´ ¯ Q ¯ r rr r r rr 1 BQ A 2 ´r 2 2 r r r r r “ Γ ´ Qrr ` pQ qrr ´ tr Q ` ´ 2 Q rr ´ Qϑϑ r Bt 6 4 3 r 1 r `Q r r `Q r rr,r zr z , rr,r rr r ` r Qrr,r r « ˜ ¸ ´ ¯ Q ¯ rϑϑ r rϑϑ 1 BQ A 2 ´r rϑϑ ` pQ r2 q ´ tr Q r2 r r ´ Q “Γ ` ´ 2 Q ϑϑ ´ Qrr ϑϑ r Bt 6 4 3 r 1r r r `Q ϑϑ,r rr r ` Qϑϑ,r r ` Qϑϑ,r zr z , r ρ « ˜ ¸ ff ´ ¯ Q r zz r r zz 1 A 1r BQ 2 2 r r r r r r “ Γ ´ Qzz ` pQ qzz ´ tr Q ` ` Qzz,rrrr ` Qzz,rr ` Qzz,rzrz , r Bt 6 4 3 r « ff ¯ ´ r rz r rz r A 1 2 Q BQ 1 2 2 r rz ` 2pQ r q ´ r Q r rz ´ r r ` 2Q r r ´ Q “Γ tr Q ` 2Q rz,r rr r ` r 2Qrz,r r rz,r zr z . rz r Bt 3 2 r r2
(16)
(17)
(18)
(19)
r “ Γ ˆ pBq0 q. We adopt the two-dimensional finite-difference method developed in our with Γ previous studies [24,25] to obtain the numerical simulation results. Here, we let the system relax from the initial boundary conditions given in Section 2.2, and the initial conditions of the upper and lower half in the bulk are specified by total rotations of ´π/2 and +π/2, respectively, consistent with the boundary conditions at the upper and lower walls given above. We consider only the equilibrium configurations that correspond to the global minimum F. In our numerical calculations, a discretization with a time step given by 10´10 is sufficient to guarantee the stability of the numerical
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procedure. In addition, our equilibration runs are 2 ˆ 106 , which is adequate for the system to reach the equilibrium. 3. Results According to parameters given in [26], a = 0.195 ˆ 106 J/m3 , B = 7.155 ˆ 106 J/m3 , C = 8.82 ˆ 106 J/m3 , L = 10.125 ˆ 10´12 J/m can be easily obtained. In our simulations, the scaled a r temperature is set to A “ 2{3, which corresponds to r s “ 1 ` 1{3. The rotational diffusion D* is set to 0.35, which is the value used in [26], and the dielectric anisotropy is set to ∆ε = 13.2 (εK = 6.5). We then obtain the exact values ξ0 = 2.64 nm and E0 = 43.1 V/um. We focus on the dependence of Ñ
structure transition on thickness (d) and then on the structure transition induced by electric field ( E ). 3.1. Structure Transition with Different Values of Thickness d and Internal Radius R1 We first analyze the equilibrium textures in the case of R1 = 30 ξ0 with different values of d. Figures 2 and 3 show the director field profile and the calculated biaxiality β2 in a cross-section along an arbitrary azimuth for different values of d. At relatively large thicknesses, the nematic system shows the defect core structure (Figure 2a,b) and is uniaxial everywhere (β2 = 0), except for a small region around the defect core (Figure 3a,b). When d decreases, the defect core clearly explodes along z (Figure 2c), and a large biaxiality propagates along z inside the system (Figure 3c). Further decrease in d results in the creation of a biaxial wall, which connects the two orthogonal uniaxial directions imposed by two cylindrical surfaces (Figures 2d and 3d) at a critical value of dc = 11.2 ξ0 . In summary, the system has transitioned from the eigenvector rotation configuration with a defect loop into the eigenvalue exchange/order reconstruction configuration by developing a thin biaxial nematic layer, which forms a cylindrical wall in the nematic system. The transition process is similar to that conducted in a planar cell in our previous studies [24,25]. Figures 2 and 3 show that the defect center is not located in the middle of the system but shifts to the internal surface; this phenomenon differs from the case of a planar cell [24,25]. Non-planar cylindrical geometry causes disclination loops to shrink, and the loop will not disappear under internal surface confinement [27]. Figure 3 shows that the deviations ∆(d) for different thicknesses (d) are 1.6 ξ0 (d = 15 ξ0 ), 1.3 ξ0 (d = 13 ξ0 ), 0.95 ξ0 (d = 11.8 ξ0 ), 0.85 ξ0 (d = 11.2 ξ0 ), 0.45 ξ0 (d = 9 ξ0 ), and 0.18 ξ0 (d = 6 ξ0 ). These finding indicate that ∆(d) decreases with decreasing d, even after eigenvalue exchange (Figure 3e,f). This behavior is due to the gradual transformation of the system to a planar structure with decreasing d. The relative deviations δ(d) = ∆(d)/d are calculated and normalized by δ(d)/δ(15 ξ0 ) to analyze the influence of non-planar geometry. Figure 4 shows the curves of δ(d)/δ(15 ξ0 ) as a function of d/ ξ0 . In the figure, the relative deviations decrease with decreasing thickness d. A clearer explanation of this phenomenon is as follows: For a planar hybrid-aligned cell (R1 Ñ8), the defect center is located in the middle of the system, with the energy on both sides of the defect equal. While for the non-planar cylindrical geometry, if the director configuration keeps the configuration of a planar cell, the energy on the inner side of the defect will be lower than that on the outer side, because that the volume on the inner side is smaller, relative to the outer side. Thus the defect center shifts to the inner side, and the director configuration of the nematic meanwhile changes, until the system reaches equilibrium. For fixed R1 , the volume differences on the two sides of the middle of the system decrease with decreasing d, thus deviations decrease with decreasing d.
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5.0 Materials 2015, 8, page–page
(a) 5.0
(b) 5.0
(a)
(b)
2.5
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(r-Rprofile )/ξ0 at the equilibrium state in a cross-section along an arbitrary Figure 2. Director field azimuth 1 0arbitrary 1 Figure 2. Director field profile at the equilibrium state in a cross-section along an azimuth inside the coaxial cylindrical system for R1 = 30 ξ0 with different thicknesses (d). The red dotted lines inside the coaxial cylindrical system for R1 = 30 ξ 0 with different thicknesses (d). The red dotted Figure 2. Director fieldofprofile at the system. equilibrium state in da =cross-section along an(d) arbitrary 13 ξ0; (c) d = 11.8 ξ0; and d = 11.2 ξ0azimuth . represent the middle the simulation (a) d = 15 ξ0; (b) lines represent the middle of the simulation system. (a) d = 15 ξ 0 ; (b) d = 13 ξ 0 ; (c) d = 11.8 ξ 0 ; and inside the coaxial cylindrical system for R1 = 30 ξ0 with different thicknesses (d). The red dotted lines (d) d = 11.2 ξ 0 . represent the middle of the simulation system. (a) d = 15 ξ0; (b) d = 13 ξ0; (c) d = 11.8 ξ0; and (d) d = 11.2 ξ0.
Figure 3. Cont.
7 Figure 3. Cont. Figure 3. Cont.
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Figure 3. Biaxiality β2 for different thicknesses (d) in a cross-section along an arbitrary azimuth Figure 3. Biaxiality β2 for different thicknesses (d) in a cross-section along an arbitrary azimuth inside inside the coaxial cylindrical system for R1 = 30 ξ0. The dotted and dashed lines represent the middle the coaxial cylindrical system for R1 = 30 ξ0 . The dotted and dashed lines represent the middle of the of Figure the simulation system the center of the (d) defect, respectively. along (a) d an = 15 ξ0; (b) azimuth d = 13 ξ0; 3. Biaxiality β2 forand different thicknesses in a cross-section arbitrary simulation system and the center of the defect, respectively. (a) d = 15 ξ0 ; (b) d = 13 ξ0 ; (c) d = 11.8 ξ0 ; theξ0coaxial R1 =(f) 30 dξ0=. The (c)inside d = 11.8 ; (d) d =cylindrical 11.2 ξ0; (e)system d = 9 ξfor 0; and 6 ξ0.dotted and dashed lines represent the middle (d) d of = 11.2 ξ0 ; (e) d = 9system ξ0 ; and (f) d = 6 ξ0 . of the defect, respectively. (a) d = 15 ξ0; (b) d = 13 ξ0; the simulation and the center (c)energy d = 11.8 ξcalculation 0; (d) d = 11.2shows ξ0; (e) d that = 9 ξ0the ; andenergy (f) d = 6in ξ0.the “inner” region of the cylindrical geometry The energy calculation showsregion, that the energy in themore “inner” region cylindrical geometry is The higher than that in the “outer” indicating that energy can of be the saved by shrinking the The shows thatreduce the energy inthat the length. “inner” region of the disclination to aincalculation smaller radius to its total In addition, the shrinking of thethe is higher thanenergy that the “outer” region, indicating more energy can becylindrical saved by geometry shrinking is higher than that in the “outer” region, indicating that more energy can be saved by shrinking the disclination a smaller radiustoalso helps decrease theIn total energythe of the system.of the disclination disclination to to a smaller radius reduce itstototal length. addition, shrinking disclination to a smaller radius to reduce its total length. In addition, the shrinking of the to a smaller radius also helps to decrease the total energy of the system.
disclination to a smaller radius1.0also helps to decrease the total energy of the system. 1.0
δ(d)/δ(15ξ0) δ(d)/δ(15ξ0)
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15 15
Figure 4. Curves of calculated δ(d)/δ(15 ξ0) as a function of d/ξ0. Figure 4. Curves of calculated δ(d)/δ(15 ξ0) as a function of d/ξ0.
Figure 4. Curves of calculated δ(d)/δ(15 ξ0 ) as a function of d/ξ0 .
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Materials 2015, 8, page–page The conditions of R1 = 20 ξ0 and R1 = 40 ξ0 are also investigated to explore the influence of Materials 2015, 8, page–page internal radius R1 on equilibrium and transition. The critical d values for different R values The conditions of R1 = 20 ξtexture 0 and R1 = 40 ξ0 are also investigated c to explore the influence1 of are shown inradius Figure The dc Rincreases with decreasing indicates The conditions R1 value = 20 ξ0of and 1and = 40transition. ξ0 are also investigated explore the influence of that 1 .toThis internal R1 5. onof equilibrium texture The critical dR c values forbehavior different R1 values the smaller the radius R1 , the biaxial transition Therefore, non-planar internal radius R1 on 5. equilibrium texture and the transition. The critical dcoccurs. values for different values are shown ininternal Figure The value of dcearlier increases with decreasing R1. This behavior indicatesRa1that the are shown Figure 5.radius The value of dearlier c increases decreasing R1.occurs. This behavior indicates that the with cylindrical system induces easy transition. The cylindrical system gradually flattens smaller theininternal R1,biaxial the the with biaxial transition Therefore, a non-planar smaller the internal radius R 1 , the earlier the biaxial transition occurs. Therefore, a non-planar cylindrical easy biaxial cylindrical system gradually increasing R1 . system Thus, induces we predict that dctransition. graduallyThe becomes equal to that of a flattens planar with cell with cylindrical system induces easy biaxial cylindrical flattens increasing 1. Thus, we predict that dctransition. gradually The becomes equalsystem to thatgradually of a planar cell with increasing R1 . R increasing RR11.. Thus, we predict 2that dc gradually becomes equal to that of a planar cell with Figure 6 shows the biaxiality β in a cross-section for different values of R1 with fixed thickness increasing FigureR61. shows the biaxiality β2 in a cross-section for different values of R1 with fixed thickness d = 11.8 ξ0 . Deviations ∆(d) are 1.252 ξ0 (R1 = 20 ξ0 ), 0.95 ξ0 (R1 = 30 ξ0 ), and 0.8 ξ0 (R1 = 40 ξ0 ); Figure shows theΔ(d) biaxiality β in forξdifferent of R 1 with thickness d = 11.8 ξ0. 6Deviations are 1.25 ξ0 a(Rcross-section 1 = 20 ξ0), 0.95 0 (R1 = 30values ξ0), and 0.8 ξ0 (Rfixed 1 = 40 ξ0); this this finding that Δ(d) ∆(d) are decreases with increasing R0 1(Ror1 =with gradual flattening ofξthe system. dfinding = 11.8 indicates ξ 0 . Deviations 1.25 ξ 0 (R 1 = 20 ξ 0 ), 0.95 ξ 30 ξ 0 ), and 0.8 ξ 0 (R 1 = 40 0); this indicates that Δ(d) decreases with increasing R1 or with gradual flattening of the system. That finding is because for for fixed d, the volume ratio on two sides the middle of the decreases indicates that Δ(d) decreases R1sides or with gradual flattening ofsystem thedecreases system. That is because fixed d, the volumewith ratioincreasing on the the two ofofthe middle of the system with That increasing R . We predict that ∆(d) gradually approaches zero when R approaches infinity. is because for fixed d, the volume ratio on the two sides of the middle of the system decreases 1 R1.We predict that Δ(d) gradually approaches zero when R1 approaches 1 with increasing infinity.
with increasing R1.We predict that Δ(d) gradually approaches zero when R1 approaches infinity. 11.5 11.5 11.4
dc /ξd0 c /ξ0
11.4 11.3
11.3 11.2 11.2 11.1 11.1 11.0 11.0
20
25
20
25
30
35
40
30 R1/ξ0
35
40
R1/ξ0
Figure 5. Critical values of dc for different internal radii (R1).
Figure 5. Critical for different differentinternal internal radii Figure 5. Criticalvalues valuesof of ddcc for radii (R1(R ). 1 ). 7.5
(a)
7.5 5.0
0.000 0.1250 0.000 0.2500 0.1250 0.3750 0.2500 0.5000 0.3750 0.6250 0.5000 0.7500 0.6250 0.8750 0.7500 1.000 0.8750 1.000
1.25ξ0
(a)
1.25ξ0
5.0 2.5 z/ξ0 z/ξ0
2.5 0.0
0.0
-2.5 -2.5 -5.0 -5.0 -7.5 -7.5 7.5 7.5 5.0
(b)
0
2
4
0
2
4
6 8 (r-R1)/ξ0 6 7.5 8 (r-R1)/ξ0 (c) 7.5
0.95ξ0
(b)
0.95ξ0
5.0
5.0
2.5
0.8ξ0
z/ξ0 z/ξ0
z/ξ0 z/ξ0
2.5
2.5 0.0
0.0
0.0
0.0
-2.5
-2.5
-2.5
-2.5
-5.0
-5.0
-5.0
-5.0
-7.5
0.8ξ0
(c)
5.0
2.5
-7.5
10 10
0
2
4
0
2
4
6 (r-R1)/ξ0 6 (r-R1)/ξ0
8
10
8
10
-7.5 -7.5
2
4
2
4
6 (r-R1)/ξ0 6 (r-R1)/ξ0
8
10
8
10
Figure 6. Biaxiality β2 in a cross-section along an arbitrary azimuth inside the coaxial cylindrical 2 in a cross-section along an arbitrary azimuth inside the coaxial cylindrical Figure system 6. forBiaxiality different βinternal radii (R1) with a certain thickness d = 11.8 ξ0. The dotted and dashed Figure 6. Biaxiality β2 internal in a cross-section alongsystem an arbitrary azimuth the coaxial cylindrical 1) with a certain thickness = 11.8 of ξinside 0. the Thedefect, dotted and dashed system for different radii lines represent the middle of the(Rsimulation and the dcenter respectively. system for different internal radii (R ) with a certain thickness d = 11.8 ξ . The dotted and dashed 1Rsimulation lines of (c) the system and the center 0of the defect, respectively. lines = 20 ξ0; (b)the R1 =middle 30 ξ0; and 1 = 40 ξ0. (a) R1 represent represent = 20middle ξ0; (b) Rof 1 =the 30 ξsimulation 0; and (c) R1 system = 40 ξ0. and9the center of the defect, respectively. (a) R1 = 20 ξ0 ; (a) R1the
(b) R1 = 30 ξ0 ; and (c) R1 = 40 ξ0 .
9
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Materials 2015, 8, page–page
3.2. 3.2. Structure Structure Transition TransitionInduced Inducedby byElectric ElectricField Field E E In In this this section, section, two two models models are are established established to to analyze analyze the the combined combined effect effect of of geometry geometry and and Ñ
Ñ
electric . Boundaries onon both coaxial cylindrical surfaces areare exchanged forfor thethe twotwo models, as electric field field EE . Boundaries both coaxial cylindrical surfaces exchanged models, shown in Figure 7. Simulations are conducted at internal radius R = 30 ξ and thickness d = 15 ξ . 1 0 0 as shown in Figure 7. Simulations are conducted at internal radius R1 = 30 ξ0 and thickness d = 15 ξ0.
Figure 7. Director profile in a cross-section along an arbitrary radius of the system for the two Figure 7. Director profile in a cross-section along an arbitrary radius of the system for the two models. models. (a) Models I and (b) II. (a) Models I and (b) II.
Figures 8 and 9 show the director field profile and the calculated biaxiality β2 in a cross-section Figures 8 and 9 show the director field profile and the calculated biaxiality β2 in a cross-section along an arbitrary azimuth as a function of r E for model I, and Figures 10 and 11 show the results for along an arbitrary azimuth as a function of E for model I, and Figures 10 and 11 show the results for model II, respectively. The disclination loops shift to the internal surface at rE = 0 , which is caused model II, respectively. The disclination loops shift to the internal surface at E “ 0, which is caused by the non-planar cylindrical geometry. The defect is pushed further toward the internal surface for by the non-planar cylindrical geometry. The defect is pushed further toward the internal surface for model II with with increasing increasing E, , whereasthe thedefect defectinitially initiallyshifts shiftstotothe themiddle middleof ofthe the system system and and then then rE whereas model to the external surface for model II. Because of the strong anchoring boundaries, the defects exhibit to the external surface for model II. Because of the strong anchoring boundaries, the defects exhibit dramaticchanges changesin in shape asdistance the distance between defectand center and the surfacedecreasing, boundary dramatic shape as the between defect center the surface boundary decreasing, especially when the lies defect lies closea biaxial to the layer surface, a biaxial layer is especially when the defect center verycenter close to thevery surface, is established, that is established, that is order reconstruction occurs [9,24,25]. Our results show that order reconstruction order reconstruction occurs [9,24,25]. Our results show that order reconstruction occurs at critical “ I and rcritical rc E = 0.21 occurs of atE values and E 0.25 for models and II, respectively. For andofE pIIq for models II, respectively. For Icomparison, a planar cell values c pIq “ 0.21 c Ι 0.25 c ΙΙ = is also simulated with the parameters, and the the corresponding reduced and critical of electric comparison, a planar cellsame is also simulated with same parameters, thevalue corresponding rc0 “ 0.24. The values are clearly ranked as follows: E rc pIq ă E rc0 ă E rc pIIq. field is E reduced critical value of electric field is E c 0 = 0.24. The values are clearly ranked as follows: Non-planar cylindrical geometry induces disclination loops to shrink for both models, whereas ( Ι ) < E Ñ E E c ( ΙΙ ) . tends to enforce the nematic director along the Ñ c c0 < E , which e r axis, expulses the defect electric field cylindrical induces disclination loops to shrink for boththat models, whereas to theNon-planar internal and external geometry surfaces for models I and II, respectively. It means the effects of Ñ electric field Egeometry , which tends to enforce theEnematic director theI, ebut expulses defect the cylindrical and electric field are common foralong model opposite for the model II. r axis,
()
( )
Hence, the transition processsurfaces differs between theItwo The common action theeffects cylindrical to the internal and external for models andmodels. II, respectively. It means thatofthe of the
Ñ
geometry electric field for model defect close to the surface boundaryfor more easily cylindricaland geometry and E electric fieldI makes common for model I, but opposite model II. E arethe Ñ Hence, the transition process differs between the two models. The common action of the cylindrical than model II, since the opposite action of the cylindrical geometry and electric field E makes it
geometry field I makes the defectFrom closethe to the surface boundaryaction more E fortomodel difficult forand theelectric defect to be close the surface boundary. above, the common
Ñ
easily than modelgeometry II, since and the opposite action of the cylindrical geometry and electric field E of the cylindrical electric field E for model I facilitates order reconstruction, while Ñ makes it difficult to be geometry close to the surface boundary. From the IIabove, the common the opposite actionfor of the thedefect cylindrical and electric field E for model complicates order
Ñ action of the cylindrical geometry andthe electric model I facilitates order reconstruction, E forfield reconstruction. For a planar cell, only actionfield of electric E exists. Considering these factors, r r r while the opposite action of the cylindrical geometry and electric field for model II complicates E our result Ec pIq ă Ec0 ă Ec pIIq is reasonable. that the radial uniform electric field has been assumed, which is difficult to orderNote reconstruction. For adirection planar cell, only the action of electric field Considering these E exists. implement almost Our results only give the qualitative analysis about Ec impossible Ι < Ec 0
Order Reconstruction in a Nanoconfined Nematic Liquid Crystal between Two Coaxial Cylinders Xuan Zhou, Zhidong Zhang *, Qian Zhang and Wenjiang Ye Received: 8 October 2015; Accepted: 16 November 2015; Published: 30 November 2015 Academic Editor: Ingo Dierking Department of Physics, Hebei University of Technology, Tianjin 300401, China; [email protected] (X.Z.); [email protected] (Q.Z.); [email protected] (W.Y.) * Correspondence: [email protected]; Tel.: +86-22-6043-5589; Fax: +86-22-6043-8055
Abstract: The dynamics of a disclination loop (s = ˘1/2) in nematic liquid crystals constrained between two coaxial cylinders were investigated based on two-dimensional Landau–de Gennes tensorial formalism by using a finite-difference iterative method. The effect of thickness (d = R2 ´ R1 , where R1 and R2 represent the internal and external radii of the cylindrical cavity, respectively) on the director distribution of the defect was simulated using different R1 values. The results show that the order reconstruction occurs at a critical value of dc , which decreases with increasing inner ratio R1 . The loop also shrinks, and the defect center deviates from the middle of the system, which is a non-planar structure. The deviation decreases with decreasing d or increasing R1 , implying that the system tends to be a planar cell. Two models were then established to analyze the combined effect of non-planar geometry and electric field. The common action of these parameters facilitates order reconstruction, whereas their opposite action complicates the process. Keywords: liquid crystal; order reconstruction; disclination loop; cylindrical wall; biaxial transition; Landau–de Gennes theory
1. Introduction The equilibrium configuration of a confined nematic liquid crystal (NLC) is caused by interplay among surface interaction, elastic distortion, and finite-size effect. Confined systems that suffer from continuous symmetry-breaking transitions often display topological defects [1–4], which are utilized in new-generation LC devices. Therefore, study of defects is an important field in physics. In particular, defects in nanoconfined LC cells have received increased research attention. A weak local perturbation in an LC confined in a nanoscale cell can stimulate an apparent mesoscopic or even macroscopic response [5]. Eigenvalue exchange/order reconstruction occurs under severe confinement in the presence of antagonistic anchorings, and this mechanism relaxes surface-induced frustrations. Order reconstruction structures have been investigated in detail [6–11] for various boundary conditions in nematic cells bounded by parallel walls, for which the characteristic linear size of the confining plates is large and virtually infinite compared with the cell thickness. These structures are stable in a sufficiently thin cell [6,7], whose gap is comparable with the order parameter length ξ0 (characteristic length for order-parameter changes), or when a sufficiently strong electric or magnetic field is applied [8,9]. Studies [12–14] revealed that mesoscopic Landau–de Gennes theory can accurately predict the structural and phase behavior of severely confined LCs. Although LC defects have been intensively studied theoretically and experimentally, LC shells have been rarely investigated. LC shells have received increased research interest because of their potential to generate colloids with a valence, which can be used to build colloidal architectures for
Materials 2015, 8, 8072–8086; doi:10.3390/ma8125446
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Materials 2015, 8, 8072–8086
photonic applications. Several scholars have numerically simulated the possible defect configurations in nematic and smectic shells [15–17]. NLCs in a cylindrical cavity have been studied for approximately 40 years. Cylindrical geometry is the most convenient geometry after planar cell geometry for theoretical and experimental analyses. Ñ As such, all experimentally observed and theoretically possible configurations of the director field n yield a complete or approximate analytic description [18]. The possible equilibrium configurations of Ñ an NLC in a cylindrical cavity primarily depend on how n is anchored to the cylinder surface. In this study, we investigated the structural behavior of a nanoscale NLC system confined in a cylindrical cavity with a topological defect loop by using the finite-difference iterative method. The outline of the paper is as follows. In Section 2, we introduce the phenomenological model employed and describe the geometry of the problem and our parametrization. The results are presented in Section 3, and the conclusions are summarized in Section 4. 2. Theoretical Basis 2.1. Free Energy Our theoretical argument is based on Landau–de Gennes theory [19], wherein the orientational order of LC is described by a second-rank symmetric and traceless tensor [3]: Q“
3 ÿ
Ñ
Ñ
λi e i b e i
(1)
i “1
Ñ
where e i is the orthogonal unit vector the eigenvector of Q, and λi is its eigenvalue. ˆ representing ˙ 1 2 The range of eigenvalues must be λi P ´ , to interpret Q as the traceless second-moment tensor 3 3 of the molecular distribution function. Q vanishes in the isotropic phase, whereas this parameter has two degenerate eigenvalues in the uniaxial ordering and can be represented by ˆ Q“S
1 nbn´ I 3
Ñ
Ñ
˙ (2)
Ñ
where n is the nematic director pointing along the local uniaxial ordering direction, and S is the uniaxial scalar parameter expressing the magnitude of fluctuations about the nematic director. In Equation (2), S can be either positive or negative. The ensemble of molecules represented by Q Ñ Ñ tends to align along n when S is positive and tends to lie in the plane orthogonal to n when S is negative. Finally, the LC is in a biaxial state when all the eigenvalues of Q are distinct. The degree of biaxiality is expressed by the parameter β2 defined as [20] “ ` ˘‰2 6 tr Q3 β “ 1 ´ “ ` ˘‰3 , tr Q2 2
(3)
This equation is a convenient parameter for illustrating spatial inhomogeneities of Q and ranges in the interval [0,1]. All uniaxial states with two degenerate eigenvalues correspond to β2 = 0, whereas states with maximal biaxiality correspond to β2 = 1. As tr(Q3 ) = 3detQ, states with β2 = 1 are those with det Q = 0, which implies that at least one eigenvalue of Q vanishes. The Landau–de Gennes free energy density of LC is given by f = fbulk {Qαβ } + felastic {Qαβ ,5} + fdielectric {Qαβ }, in which ¯2 A B C´ (4) f bulk “ trQ2 ´ trQ3 ` trQ2 2 3 4
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is the bulk energy that describes a homogeneous phase. In fbulk , B and C are constants, and A is assumed to vary with temperature T in the form of A = a (T ´ T*), where a is a positive constant and T* is the nematic supercooling temperature. Equation (4) the bulk ˜ provides ¸ equilibrium value of c 24AC B 1+ 1 ´ , which depends on the the uniaxial scalar order parameter in Equation (2) Seq = 4C B2 temperature. The free-energy felastic , which penalizes gradients in the tensor order parameter field, is given in the form 1 f elastic “ L |∇ Q|2 (5) 2 where the one-elastic-constant approximation is used for simplicity, and the elasticity of the system is represented by a single positive elastic constant L. The dielectric contribution to the free energy density is given by [21] f dielectric “ ´
ε0 Ñ Ñ E ¨ εE 2
(6)
where ε0 is the vacuum electric permeability constant, and ε is the dielectric tensor to describe the Ñ
local anisotropic response of the nematic ordering to E . ε is generally expressed as ε “ εi I ` ε a Q
(7)
where εi = (εk + 2εK )/3 and εa = (εk ´ εK )/Seq are isotropic and anisotropic dielectric susceptibilities, Ñ
respectively. When εa > 0, similar to the following calculation, nematic uniaxial ordering is along E and therefore fdielectric can be rewritten as f dielectric “ ´
ε0 2
ˆ
Ñ2
Ñ
Ñ
˙
εi E ` ε a E ¨ Q E
(8)
2.2. Geometry of the Problem Let us consider an NLC confined between two coaxial cylindrical surfaces whose internal and external radii are R1 and R2 , and the thickness of the shell is d = R2 ´ ´ R1 (Figure 1a). ¯ The standard polar cylindrical coordinates pr, ϑ, zq and the corresponding local frame Ñ
Ñ Ñ Ñ e r , e ϑ, e z
are introduced,
Ñ
where e z points along the symmetry axis, e r is the radial unit vector emanating from the symmetry Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
axis, and e ϑ :“ e z ˆ e r . The electric field E is applied along the e r axis direction, i.e., E “ E e r , Ñ which promotes the alignment of the nematic director along the e r axis. LC molecules on the surfaces of internal and external cylinders are strongly anchored along the parallel and perpendicular directions, respectively. According to the investigations of Cavallaro et al. [22], we make the assumption that the LC texture exhibits a cylindrical symmetry along the cylindrical axis, i.e., the nematic orientation is independent of ϑ. By varying the polar angle Ñ θ1 on the internal surface, where θ1 is the angle with respect to e z in the plane ϑ-z, we have verified that the choice of θ1 = 0 has the minimum energy, i.e., the director field has no azimuthal component and occurs in the plane r-z. This is a realistic choice of boundary condition for a cylindrical surface in industrial production. Generally, a homogenous planar alignment is achieved firstly by rubbing on flexible substrate, and then the substrate is fit on the solid cylinder, with the rubbing direction along the symmetric axis. It follows that the texture can be discussed in terms of 2D nematics corresponding Ñ to each radial slice. In the plane r-z, the nematic director is given by n “ psinβ, 0, cosβq, where â is Ñ the angle with respect to e z (as shown in Figure 1) and the order tensor Q can be represented as
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n = ( sin β,0,cos β) , where â is the angle with respect to ez
(as shown in Figure 1) and the order
tensor Q can be represented as Materials 2015, 8, 8072–8086
Hence,
eϑ
0 Qrz Qrr Q ( r , z ) = 0 Qϑϑ 0 , ¨ ˛ Q Q 0 Q 0 Q rr rz zz ‹ ˚ rz
(9)
Q pr, zq “ ˝
(9) 0 Qϑϑ 0 ‚, Q 0 Q zz is always an eigenvector of Q.rzThis configuration rules out distortions twisted
Ñ always an(9)eigenvector of Q.departures This configuration out distortions eϑHence, e1 , e2twisted , e3 ) of . Qrz einϑ isEquation reveals the of the rules eigenframe ( ´ ¯ along Ñe ϑ. Qrz in Equation (9) reveals the departures of the eigenframe Ñe 1 , Ñe 2 , Ñe 3 of Q from er , eÑϑ , eÑz )¯, with both frames coinciding when Qrz = 0. from´(Ñ
along
Q
e r , e ϑ , e z , with both frames coinciding when Qrz = 0.
We denote the free boundary conditions at the upper and lower lateral walls z = ±dz/2, with We denote the free boundary conditions at the upper and lower lateral walls z = ˘dz /2, with initialinitial uniaxial ordering −π/2(z(z dz/2)and and +π/2 = z−d z/2). These boundary uniaxial orderingby bytotal total rotations rotations ofof´π/2 = d=z /2) +π/2 (z =(z´d /2). These boundary conditions are are consistent with of aa defect defectloop loopwith with topological charge s = −1/2. conditions consistent withthe thegeneration generation of topological charge s = ´1/2. Figure 1b shows the director profile in a cross-section along an arbitrary radius of the system. Figure 1b shows the director profile in a cross-section along an arbitrary radius of the system. The Theare results also applicable to defect the defect loop withtopological topological charge with exchanged results alsoare applicable to the loop with charges = s 1/2, = 1/2, with exchanged conditions of the upper and lower walls. conditions of the upper and lower walls.
Figure 1. Geometry of the problem. cavityfilled filledwith with NLCs; (b) the Figure 1. Geometry of the problem.(a)(a)Sketch Sketchof of cylindrical cylindrical cavity NLCs; andand (b) the director profile in a cross-section along an arbitrary radius of the system. director profile in a cross-section along an arbitrary radius of the system.
2.3. Scaling and and Dimensionless Evolution 2.3. Scaling Dimensionless EvolutionEquations Equations «
ff
4 r ijB” Qij {q0 ,r introducedthe the following quantities: fr ” f { f ≡ f3 , Q We We introduced followingdimensionless dimensionless quantities: , Qzij ”≡z{ξ Qij0 , q0 , p4Cq 3
B4
b (4C ) r i ε0 , E r “ E{E0 , where q0 “ B{4C is the superheating 2L{ Aε c q = B 4C z ≡ z ξ0 , r ≡ r ξ0 , ε a = εa q0 εi , E0 = q0 ξ0 2 L A εi ε 0 , E = E dE0L, where 4CL0 order parameter at the nematic superheating temperature T**, and ξ0 “ “ is the B2 T**, and is the superheating order parameter at the nematic superheating Bq temperature 0 characteristic length for order-parameter changes. Equations (4), (5), and (8) can be reduced to r r ” r{ξ0 , r εa “ εa q0 {εi , E0 “ pq0 {ξ0 q
ξ0 =
L = Bq0
(
)
4CL is the characteristic length for order-parameter changes. Equations (4), (5), r B2 A 1 ´ r2 ¯2 r r2 1 r3
and (8) can be reduced to
tr Q ´ tr Q ` tr Q 3 16 1 ˇˇ r rˇˇ2 A frelastic “ 1 2 ˇ∇ 3Q+ˇ 1 trQ 2 = trQ 2 − trQ ı 12 16 13” r2 r rr E r2 E `r εa Q frdielectric “ ´ r A
f bulk “
fbulk
(10)
12
(
1 felastic = ∇ Q 2
4
2
(11)
(10)
(12)
2
1 fdielectric = − E 2 + ε a Q rr E 2 A 8075
)
(11)
(12)
Materials 2015, 8, 8072–8086
r “ 24AC defines the temperature scale. Isotropic–nematic transition where the reduced parameter A B2 r “ 8{9. In polar cylindrical coordinates, Equation (11) can be expressed precisely by occurs at A »˜
fre “
¸2 ˜ ¸2 ˜ ¸2 ˜ ¸2 ˜ ¸2 ˜ ¸2 fi r r r r r r 1 – B Qrr B Qrr B Qϑϑ B Qϑϑ B Qzz B Qzz fl ` ` ` ` ` 2 Br r Br z Br r Br z Br r Br z »˜ ¸2 ˜ ¸2 fi ´ ¯ r rz r rz BQ BQ r2 ` Q r2 ` Q r2 ´ 2Q r rr Q rϑϑ . fl ` 1 Q ` `– rr rz ϑϑ r Br r Br z r2
(13)
Thus, the reduced uniaxial ordering has the form 2
Ñ r “ Sp3 r Ñ Q n b n ´ Iq ,
(14)
a Seq r is the reduced uniaxial scalar parameter at equilibrium. “ 1` 1´ A q0 We compute the evolution of LC with dynamic theory for tensor order-parameter field Q(r,z,t). Ñ The local values of the scalar-order parameter S and the director n can be calculated from Q by using the highest eigenvalue and the associated eigenvector, respectively. According to [23], the evolution equation describing the dynamics of Q can be written as where r s“
„ ˆ ˙ δf 1 δf BQ “Γ ´ ` tr I , Bt δQ 3 δQ
(15)
where Γ = 6D*/[1 ´ 3tr(Q2 )]2 , D* is the rotational diffusion for the nematic, and δ f {δQ is assumed to be symmetrical. Numerical calculations are performed using the reduced variables. When the functional r can be obtained. derivatives in Equation (15) are evaluated, the following evolution equations for Q « ˜ ¸ ´ ¯ Q ¯ r rr r r rr 1 BQ A 2 ´r 2 2 r r r r r “ Γ ´ Qrr ` pQ qrr ´ tr Q ` ´ 2 Q rr ´ Qϑϑ r Bt 6 4 3 r 1 r `Q r r `Q r rr,r zr z , rr,r rr r ` r Qrr,r r « ˜ ¸ ´ ¯ Q ¯ rϑϑ r rϑϑ 1 BQ A 2 ´r rϑϑ ` pQ r2 q ´ tr Q r2 r r ´ Q “Γ ` ´ 2 Q ϑϑ ´ Qrr ϑϑ r Bt 6 4 3 r 1r r r `Q ϑϑ,r rr r ` Qϑϑ,r r ` Qϑϑ,r zr z , r ρ « ˜ ¸ ff ´ ¯ Q r zz r r zz 1 A 1r BQ 2 2 r r r r r r “ Γ ´ Qzz ` pQ qzz ´ tr Q ` ` Qzz,rrrr ` Qzz,rr ` Qzz,rzrz , r Bt 6 4 3 r « ff ¯ ´ r rz r rz r A 1 2 Q BQ 1 2 2 r rz ` 2pQ r q ´ r Q r rz ´ r r ` 2Q r r ´ Q “Γ tr Q ` 2Q rz,r rr r ` r 2Qrz,r r rz,r zr z . rz r Bt 3 2 r r2
(16)
(17)
(18)
(19)
r “ Γ ˆ pBq0 q. We adopt the two-dimensional finite-difference method developed in our with Γ previous studies [24,25] to obtain the numerical simulation results. Here, we let the system relax from the initial boundary conditions given in Section 2.2, and the initial conditions of the upper and lower half in the bulk are specified by total rotations of ´π/2 and +π/2, respectively, consistent with the boundary conditions at the upper and lower walls given above. We consider only the equilibrium configurations that correspond to the global minimum F. In our numerical calculations, a discretization with a time step given by 10´10 is sufficient to guarantee the stability of the numerical
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procedure. In addition, our equilibration runs are 2 ˆ 106 , which is adequate for the system to reach the equilibrium. 3. Results According to parameters given in [26], a = 0.195 ˆ 106 J/m3 , B = 7.155 ˆ 106 J/m3 , C = 8.82 ˆ 106 J/m3 , L = 10.125 ˆ 10´12 J/m can be easily obtained. In our simulations, the scaled a r temperature is set to A “ 2{3, which corresponds to r s “ 1 ` 1{3. The rotational diffusion D* is set to 0.35, which is the value used in [26], and the dielectric anisotropy is set to ∆ε = 13.2 (εK = 6.5). We then obtain the exact values ξ0 = 2.64 nm and E0 = 43.1 V/um. We focus on the dependence of Ñ
structure transition on thickness (d) and then on the structure transition induced by electric field ( E ). 3.1. Structure Transition with Different Values of Thickness d and Internal Radius R1 We first analyze the equilibrium textures in the case of R1 = 30 ξ0 with different values of d. Figures 2 and 3 show the director field profile and the calculated biaxiality β2 in a cross-section along an arbitrary azimuth for different values of d. At relatively large thicknesses, the nematic system shows the defect core structure (Figure 2a,b) and is uniaxial everywhere (β2 = 0), except for a small region around the defect core (Figure 3a,b). When d decreases, the defect core clearly explodes along z (Figure 2c), and a large biaxiality propagates along z inside the system (Figure 3c). Further decrease in d results in the creation of a biaxial wall, which connects the two orthogonal uniaxial directions imposed by two cylindrical surfaces (Figures 2d and 3d) at a critical value of dc = 11.2 ξ0 . In summary, the system has transitioned from the eigenvector rotation configuration with a defect loop into the eigenvalue exchange/order reconstruction configuration by developing a thin biaxial nematic layer, which forms a cylindrical wall in the nematic system. The transition process is similar to that conducted in a planar cell in our previous studies [24,25]. Figures 2 and 3 show that the defect center is not located in the middle of the system but shifts to the internal surface; this phenomenon differs from the case of a planar cell [24,25]. Non-planar cylindrical geometry causes disclination loops to shrink, and the loop will not disappear under internal surface confinement [27]. Figure 3 shows that the deviations ∆(d) for different thicknesses (d) are 1.6 ξ0 (d = 15 ξ0 ), 1.3 ξ0 (d = 13 ξ0 ), 0.95 ξ0 (d = 11.8 ξ0 ), 0.85 ξ0 (d = 11.2 ξ0 ), 0.45 ξ0 (d = 9 ξ0 ), and 0.18 ξ0 (d = 6 ξ0 ). These finding indicate that ∆(d) decreases with decreasing d, even after eigenvalue exchange (Figure 3e,f). This behavior is due to the gradual transformation of the system to a planar structure with decreasing d. The relative deviations δ(d) = ∆(d)/d are calculated and normalized by δ(d)/δ(15 ξ0 ) to analyze the influence of non-planar geometry. Figure 4 shows the curves of δ(d)/δ(15 ξ0 ) as a function of d/ ξ0 . In the figure, the relative deviations decrease with decreasing thickness d. A clearer explanation of this phenomenon is as follows: For a planar hybrid-aligned cell (R1 Ñ8), the defect center is located in the middle of the system, with the energy on both sides of the defect equal. While for the non-planar cylindrical geometry, if the director configuration keeps the configuration of a planar cell, the energy on the inner side of the defect will be lower than that on the outer side, because that the volume on the inner side is smaller, relative to the outer side. Thus the defect center shifts to the inner side, and the director configuration of the nematic meanwhile changes, until the system reaches equilibrium. For fixed R1 , the volume differences on the two sides of the middle of the system decrease with decreasing d, thus deviations decrease with decreasing d.
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(r-Rprofile )/ξ0 at the equilibrium state in a cross-section along an arbitrary Figure 2. Director field azimuth 1 0arbitrary 1 Figure 2. Director field profile at the equilibrium state in a cross-section along an azimuth inside the coaxial cylindrical system for R1 = 30 ξ0 with different thicknesses (d). The red dotted lines inside the coaxial cylindrical system for R1 = 30 ξ 0 with different thicknesses (d). The red dotted Figure 2. Director fieldofprofile at the system. equilibrium state in da =cross-section along an(d) arbitrary 13 ξ0; (c) d = 11.8 ξ0; and d = 11.2 ξ0azimuth . represent the middle the simulation (a) d = 15 ξ0; (b) lines represent the middle of the simulation system. (a) d = 15 ξ 0 ; (b) d = 13 ξ 0 ; (c) d = 11.8 ξ 0 ; and inside the coaxial cylindrical system for R1 = 30 ξ0 with different thicknesses (d). The red dotted lines (d) d = 11.2 ξ 0 . represent the middle of the simulation system. (a) d = 15 ξ0; (b) d = 13 ξ0; (c) d = 11.8 ξ0; and (d) d = 11.2 ξ0.
Figure 3. Cont.
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Figure 3. Biaxiality β2 for different thicknesses (d) in a cross-section along an arbitrary azimuth Figure 3. Biaxiality β2 for different thicknesses (d) in a cross-section along an arbitrary azimuth inside inside the coaxial cylindrical system for R1 = 30 ξ0. The dotted and dashed lines represent the middle the coaxial cylindrical system for R1 = 30 ξ0 . The dotted and dashed lines represent the middle of the of Figure the simulation system the center of the (d) defect, respectively. along (a) d an = 15 ξ0; (b) azimuth d = 13 ξ0; 3. Biaxiality β2 forand different thicknesses in a cross-section arbitrary simulation system and the center of the defect, respectively. (a) d = 15 ξ0 ; (b) d = 13 ξ0 ; (c) d = 11.8 ξ0 ; theξ0coaxial R1 =(f) 30 dξ0=. The (c)inside d = 11.8 ; (d) d =cylindrical 11.2 ξ0; (e)system d = 9 ξfor 0; and 6 ξ0.dotted and dashed lines represent the middle (d) d of = 11.2 ξ0 ; (e) d = 9system ξ0 ; and (f) d = 6 ξ0 . of the defect, respectively. (a) d = 15 ξ0; (b) d = 13 ξ0; the simulation and the center (c)energy d = 11.8 ξcalculation 0; (d) d = 11.2shows ξ0; (e) d that = 9 ξ0the ; andenergy (f) d = 6in ξ0.the “inner” region of the cylindrical geometry The energy calculation showsregion, that the energy in themore “inner” region cylindrical geometry is The higher than that in the “outer” indicating that energy can of be the saved by shrinking the The shows thatreduce the energy inthat the length. “inner” region of the disclination to aincalculation smaller radius to its total In addition, the shrinking of thethe is higher thanenergy that the “outer” region, indicating more energy can becylindrical saved by geometry shrinking is higher than that in the “outer” region, indicating that more energy can be saved by shrinking the disclination a smaller radiustoalso helps decrease theIn total energythe of the system.of the disclination disclination to to a smaller radius reduce itstototal length. addition, shrinking disclination to a smaller radius to reduce its total length. In addition, the shrinking of the to a smaller radius also helps to decrease the total energy of the system.
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Figure 4. Curves of calculated δ(d)/δ(15 ξ0 ) as a function of d/ξ0 .
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Materials 2015, 8, page–page The conditions of R1 = 20 ξ0 and R1 = 40 ξ0 are also investigated to explore the influence of Materials 2015, 8, page–page internal radius R1 on equilibrium and transition. The critical d values for different R values The conditions of R1 = 20 ξtexture 0 and R1 = 40 ξ0 are also investigated c to explore the influence1 of are shown inradius Figure The dc Rincreases with decreasing indicates The conditions R1 value = 20 ξ0of and 1and = 40transition. ξ0 are also investigated explore the influence of that 1 .toThis internal R1 5. onof equilibrium texture The critical dR c values forbehavior different R1 values the smaller the radius R1 , the biaxial transition Therefore, non-planar internal radius R1 on 5. equilibrium texture and the transition. The critical dcoccurs. values for different values are shown ininternal Figure The value of dcearlier increases with decreasing R1. This behavior indicatesRa1that the are shown Figure 5.radius The value of dearlier c increases decreasing R1.occurs. This behavior indicates that the with cylindrical system induces easy transition. The cylindrical system gradually flattens smaller theininternal R1,biaxial the the with biaxial transition Therefore, a non-planar smaller the internal radius R 1 , the earlier the biaxial transition occurs. Therefore, a non-planar cylindrical easy biaxial cylindrical system gradually increasing R1 . system Thus, induces we predict that dctransition. graduallyThe becomes equal to that of a flattens planar with cell with cylindrical system induces easy biaxial cylindrical flattens increasing 1. Thus, we predict that dctransition. gradually The becomes equalsystem to thatgradually of a planar cell with increasing R1 . R increasing RR11.. Thus, we predict 2that dc gradually becomes equal to that of a planar cell with Figure 6 shows the biaxiality β in a cross-section for different values of R1 with fixed thickness increasing FigureR61. shows the biaxiality β2 in a cross-section for different values of R1 with fixed thickness d = 11.8 ξ0 . Deviations ∆(d) are 1.252 ξ0 (R1 = 20 ξ0 ), 0.95 ξ0 (R1 = 30 ξ0 ), and 0.8 ξ0 (R1 = 40 ξ0 ); Figure shows theΔ(d) biaxiality β in forξdifferent of R 1 with thickness d = 11.8 ξ0. 6Deviations are 1.25 ξ0 a(Rcross-section 1 = 20 ξ0), 0.95 0 (R1 = 30values ξ0), and 0.8 ξ0 (Rfixed 1 = 40 ξ0); this this finding that Δ(d) ∆(d) are decreases with increasing R0 1(Ror1 =with gradual flattening ofξthe system. dfinding = 11.8 indicates ξ 0 . Deviations 1.25 ξ 0 (R 1 = 20 ξ 0 ), 0.95 ξ 30 ξ 0 ), and 0.8 ξ 0 (R 1 = 40 0); this indicates that Δ(d) decreases with increasing R1 or with gradual flattening of the system. That finding is because for for fixed d, the volume ratio on two sides the middle of the decreases indicates that Δ(d) decreases R1sides or with gradual flattening ofsystem thedecreases system. That is because fixed d, the volumewith ratioincreasing on the the two ofofthe middle of the system with That increasing R . We predict that ∆(d) gradually approaches zero when R approaches infinity. is because for fixed d, the volume ratio on the two sides of the middle of the system decreases 1 R1.We predict that Δ(d) gradually approaches zero when R1 approaches 1 with increasing infinity.
with increasing R1.We predict that Δ(d) gradually approaches zero when R1 approaches infinity. 11.5 11.5 11.4
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3.2. 3.2. Structure Structure Transition TransitionInduced Inducedby byElectric ElectricField Field E E In In this this section, section, two two models models are are established established to to analyze analyze the the combined combined effect effect of of geometry geometry and and Ñ
Ñ
electric . Boundaries onon both coaxial cylindrical surfaces areare exchanged forfor thethe twotwo models, as electric field field EE . Boundaries both coaxial cylindrical surfaces exchanged models, shown in Figure 7. Simulations are conducted at internal radius R = 30 ξ and thickness d = 15 ξ . 1 0 0 as shown in Figure 7. Simulations are conducted at internal radius R1 = 30 ξ0 and thickness d = 15 ξ0.
Figure 7. Director profile in a cross-section along an arbitrary radius of the system for the two Figure 7. Director profile in a cross-section along an arbitrary radius of the system for the two models. models. (a) Models I and (b) II. (a) Models I and (b) II.
Figures 8 and 9 show the director field profile and the calculated biaxiality β2 in a cross-section Figures 8 and 9 show the director field profile and the calculated biaxiality β2 in a cross-section along an arbitrary azimuth as a function of r E for model I, and Figures 10 and 11 show the results for along an arbitrary azimuth as a function of E for model I, and Figures 10 and 11 show the results for model II, respectively. The disclination loops shift to the internal surface at rE = 0 , which is caused model II, respectively. The disclination loops shift to the internal surface at E “ 0, which is caused by the non-planar cylindrical geometry. The defect is pushed further toward the internal surface for by the non-planar cylindrical geometry. The defect is pushed further toward the internal surface for model II with with increasing increasing E, , whereasthe thedefect defectinitially initiallyshifts shiftstotothe themiddle middleof ofthe the system system and and then then rE whereas model to the external surface for model II. Because of the strong anchoring boundaries, the defects exhibit to the external surface for model II. Because of the strong anchoring boundaries, the defects exhibit dramaticchanges changesin in shape asdistance the distance between defectand center and the surfacedecreasing, boundary dramatic shape as the between defect center the surface boundary decreasing, especially when the lies defect lies closea biaxial to the layer surface, a biaxial layer is especially when the defect center verycenter close to thevery surface, is established, that is established, that is order reconstruction occurs [9,24,25]. Our results show that order reconstruction order reconstruction occurs [9,24,25]. Our results show that order reconstruction occurs at critical “ I and rcritical rc E = 0.21 occurs of atE values and E 0.25 for models and II, respectively. For andofE pIIq for models II, respectively. For Icomparison, a planar cell values c pIq “ 0.21 c Ι 0.25 c ΙΙ = is also simulated with the parameters, and the the corresponding reduced and critical of electric comparison, a planar cellsame is also simulated with same parameters, thevalue corresponding rc0 “ 0.24. The values are clearly ranked as follows: E rc pIq ă E rc0 ă E rc pIIq. field is E reduced critical value of electric field is E c 0 = 0.24. The values are clearly ranked as follows: Non-planar cylindrical geometry induces disclination loops to shrink for both models, whereas ( Ι ) < E Ñ E E c ( ΙΙ ) . tends to enforce the nematic director along the Ñ c c0 < E , which e r axis, expulses the defect electric field cylindrical induces disclination loops to shrink for boththat models, whereas to theNon-planar internal and external geometry surfaces for models I and II, respectively. It means the effects of Ñ electric field Egeometry , which tends to enforce theEnematic director theI, ebut expulses defect the cylindrical and electric field are common foralong model opposite for the model II. r axis,
()
( )
Hence, the transition processsurfaces differs between theItwo The common action theeffects cylindrical to the internal and external for models andmodels. II, respectively. It means thatofthe of the
Ñ
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geometry field I makes the defectFrom closethe to the surface boundaryaction more E fortomodel difficult forand theelectric defect to be close the surface boundary. above, the common
Ñ
easily than modelgeometry II, since and the opposite action of the cylindrical geometry and electric field E of the cylindrical electric field E for model I facilitates order reconstruction, while Ñ makes it difficult to be geometry close to the surface boundary. From the IIabove, the common the opposite actionfor of the thedefect cylindrical and electric field E for model complicates order
Ñ action of the cylindrical geometry andthe electric model I facilitates order reconstruction, E forfield reconstruction. For a planar cell, only actionfield of electric E exists. Considering these factors, r r r while the opposite action of the cylindrical geometry and electric field for model II complicates E our result Ec pIq ă Ec0 ă Ec pIIq is reasonable. that the radial uniform electric field has been assumed, which is difficult to orderNote reconstruction. For adirection planar cell, only the action of electric field Considering these E exists. implement almost Our results only give the qualitative analysis about Ec impossible Ι < Ec 0
